We study a linear quadratic optimal control problem with stochastic coefficients and a terminal state constraint, which may be in force merely on a set with positive, but not necessarily full, probability. Under such a partial terminal constraint, the usual approach via a coupled system of a backward stochastic Riccati equation and a linear backward equation breaks down. As a remedy, we introduce a family of auxiliary problems parametrized by the supersolutions to this Riccati equation alone. The target functional of these problems dominates the original constrained one and allows for an explicit description of both the optimal control policy and the auxiliary problem's value in terms of a suitably constructed optimal signal process. This suggests that, for the minimal supersolution of the Riccati equation, the minimizers of the auxiliary problem coincide with those of the original problem, a conjecture that we see confirmed in all examples understood so far.